# Optimal control of a non-smooth semilinear elliptic equation

arXiv: https://arxiv.org/abs/1705.00939 · HTML: https://mlsanity.com/paper/1705.00939/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 May 2017.

## Built on by (most cited first, 1 of 1)

- [On the Identification and Optimization of Nonsmooth Superposition Operators in Semilinear Elliptic PDEs](https://mlsanity.com/paper/2306.05185/) (2023)

## Abstract (arXiv)

This paper is concerned with an optimal control problem governed by a non-smooth semilinear elliptic equation. We show that the control-to-state mapping is directionally differentiable and precisely characterize its Bouligand subdifferential. By means of a suitable regularization, first-order optimality conditions including an adjoint equation are derived and afterwards interpreted in light of the previously obtained characterization. In addition, the directional derivative of the control-to-state mapping is used to establish strong stationarity conditions. While the latter conditions are shown to be stronger, we demonstrate by numerical examples that the former conditions are amenable to numerical solution using a semi-smooth Newton method.